Let’s explore the fundamental properties and compare two essential graphs: $y = \log_2 x$ and $y = \log_{1/2} x$.
These rules are universally true for any valid base $a$ ($a>0, a \ne 1$).
The definition of a logarithm stems from its relationship with exponents:
This property shows that the base $a$ and the logarithm of base $a$ effectively undo each other.
If two logarithms with the same base are equal, their arguments must also be equal:
This is the property most often used to solve logarithmic equations.
When the base $b$ is greater than 1 ($b > 1$), the function is increasing (the graph goes up as you move from left to right).
| x Value | y=log2x | Point (x,y) |
| $\frac{1}{4}$ | -2 | ($\frac{1}{4}$, -2) |
| 1 | 0 | (1, 0) |
| 2 | 1 | (2, 1) |
| 4 | 2 | (4, 2) |
Characteristics of $y = \log_2 x$:
Behavior: Increasing function.
X-Intercept: Passes through (1, 0).
Vertical Asymptote: The line $x=0$ (the $Y$-axis).
Domain: $x > 0$.
Range: $(-\infty, \infty)$.
When the base $b$ is between 0 and 1 ($0 < b < 1$), the function is decreasing (the graph goes down as you move from left to right).
| x Value | y=log1/2x | Point (x,y) |
| $\frac{1}{4}$ | 2 | ($\frac{1}{4}$, 2) |
| 1 | 0 | (1, 0) |
| 2 | -1 | (2, -1) |
| 4 | -2 | (4, -2) |
Characteristics of $y = \log_{1/2} x$:
Behavior: Decreasing function.
X-Intercept: Passes through (1, 0).
Vertical Asymptote: The line $x=0$ (the $Y$-axis).
Domain: $x > 0$.
Range: $(-\infty, \infty)$.
The two graphs are reflections of each other across the x-axis.
$y = \log_{1/2} x$ is equivalent to $y = \log_{2^{-1}} x$, which simplifies to $y = – \log_2 x$.
Whether the function rises or falls, they both share the critical point (1, 0) and the vertical asymptote at $x=0$. The base is what determines the direction of the curve!